7. A function f : D CR -> R is called (sequentially) lower semicontinuous at a point I C D if we have f(x) R is lower semicontinuous in every z ( [a, b], then f attains its minimum value in [a, b]. Give the full proof, you cannot use the min-mar property for upper semicontinuous or continuous func- tions 8. Prove the "Banach Fixed Point Theorem": Assume that f : R -> R is a function that for some L e [0, 1) satisfies If(x) - f(y)| co In satisfies f(x) = r. (c) To show that there exist exactly one solution = = f(x) assume there are two solutions z and y and compute If (x) - f(y)|- 9. Let f : A C R -> R be uniformly continuous. (a) Show that if A is a nonempty bounded set, then f is a bounded function on A, i.c. supAf -00. (b) Show that the conclusion is false if A is not bounded. 10. Let A and B be subsets of R. Suppose that f : B -> R and g : A > R such that g(A) C B. If f and g are both uniformly continuous, will fog be uniformly continuous